. Article .

SCIENCE CHINAPhysics, Mechanics & Astronomy

November 2014 Vol. 57 No. 11: 2092–2097doi: 10.1007/s11433-014-5591-1

c© Science China Press and Springer-Verlag Berlin Heidelberg 2014 phys.scichina.com link.springer.com

Photonic phase transition in circuit quantum electrodynamicslattices coupled to superconducting phase qubits

LIU YiMin1*, JIN WuYin2 & YOU JiaBin3

1Department of Physics, Shaoguan University, Shaoguan 512005, China;2School of Mechanical & Electronical Engineering, Lanzhou University of Technology, Lanzhou 730050, China;

3Center for Quantum Technologies and Physics Department, National University of Singapore, 3 Science Drive 2, 117543, Singapore

Received May 29, 2014; accepted July 31, 2014; published online September 2, 2014

A hybrid quantum architecture was proposed to engineer a localization-delocalization phase transition of light in a two-dimensionsquare lattices of superconducting coplanar waveguide resonators, which are interconnected by current-biased Josephson junctionphase qubits. We find that the competition between the on-site repulsion and the nonlocal photonic hopping leads to the Mottinsulator-superfluid transition. By using the mean-field approach and the quantum master equation, the phase boundary betweenthese two different phases could be obtained when the dissipative effects of superconducting resonators and phase qubit are consid-ered. The good tunability of the effective on-site repulsion and photon-hopping strengths enable quantum simulation on condensedmatter physics and many-body models using such a superconducting resonator lattice system. The experimental feasibility isdiscussed using the currently available technology in the circuit QED.

circuit QED, quantum phase transition, hybrid quantum system

PACS number(s): 03.67.Lx, 03.67.Mn, 42.50.Dv, 42.50.Pq

Citation: Liu Y M, Jin W Y, You J B. Photonic phase transition in circuit quantum electrodynamics lattices coupled to superconducting phase qubits. SciChina-Phys Mech Astron, 2014, 57: 2092–2097, doi: 10.1007/s11433-014-5591-1

1 Introduction

Recently, the investigation of quantum simulator on the pho-tonic many-body physics has made great advances in a va-riety of quantum systems [1–8]. Here, the general goal isto simulate the properties of non-trivial condensed-matterphysics and strongly interacting many-body models in a well-controllable way [9]. Furthermore, it is intriguing to employ awell-controllable quantum system [10,11] (it can provide ad-ditional experimental control and permit novel measurementaccess) with a tunable Hamiltonian, such as the artificial-ly hybrid devices, to simulate the physics of another quan-tum system of interest. For example, by using ideas fromcavity quantum electrodynamics (QED), a general advantageof superconducting circuit QED [12–21], includes numer-ous different coupling mechanisms and the unique merits of

*Corresponding author (email: lym84111766@163.com)

scalability that can be implemented on a chip. In particu-lar, arrays of superconducting coplanar waveguide resonators(CPW) with tunable coupling strengths between individu-al resonators have provided the possibility for studying thestrongly-correlated states of light via the photonic process-es, and these hybrid quantum systems have become an excel-lent platform to realize non-equilibrium quantum simulation,due to their flexibility and suitability in fabrication [22–25].Moreover, because the ‘particles’ in circuit QED being sim-ulated are just circuit excitation, the particle number is notnecessarily conserved. On the other hand, unavoidable pho-ton loss, coupled with the ease of feeding in additional mi-crowave photons through the external continuous driving,makes such resonators lattices open quantum systems, whichcan also be studied in a non-equilibrium steady state. Moreimportantly, with the recent progress in the multiresonatorexperiments [26,27], our proposed architecture may serve

Liu Y M, et al. Sci China-Phys Mech Astron November (2014) Vol. 57 No. 11 2093

as a guide to becoming experimentally observable quantumphase transition (QPT) based on a large-scale CPW array andit complements the repertoire of many-body physics. Thiscould pave the way for solid-state quantum simulation andcould open interesting and new perspectives for correlatedpolariton systems. Experimentally, Underwood et al. [26]recently experimentally fabricated 25 arrays of capacitivelycoupled CPWs and demonstrated the feasibility of quantumsimulation in circuit QED system. Lucero et al. [27] exper-imentally characterized a complex circuit composed of fourphase qubits and five CPWs to realize the intricate quantumalgorithms. The above-mentioned progression renders theCPW lattices a good backbone for the study of condensed-matter physics with photons and makes our scheme morepractical.

In this paper we investigate the strongly correlated effectsin a hybrid solid-state system, which is a two-dimension (2D)square lattice of coupled CPW, where each current-biasedJosephson junction phase qubit (CBJJ) is placed at the antin-odes of the electric field of CPW [28–31]. The main moti-vation for building such a hybrid system is to combine twoadvantages: the in situ tunability of the parameters of thecircuit elements [16], individual addressing, spectroscopictechnology of measurement, and the scalability of CPW ar-rays [13,25,32,33]. Our numerical results show that the inter-play between CPW-CBJJ interaction and nonlocal tunnelingbetween adjacent CPWs leads to a localization-delocalizationtransition of light, and the Mott insulator phase (MIP) and thesuperfluid phase (SFP) can appear subsequently through con-trolling the tunable experimental parameters, such as, the on-site emitter-field interaction (namely, CPW-CBJJ couplingstrength) and nonlocal photon-photon hopping (PH) rate canbe modulated independently through the external parameters,such as the junction capacitance CJ of the CBJJ and thecoupling capacitors Cc of the CPW. In our work, the phaseboundary in a complete parameter space between MIP andSFP in the presence of dissipation can be obtained using themean-field approach and quantum master equation. Addi-tionally, the possibility of observing the QPT has been ana-lyzed by employing experimentally accessible parameters.

As shown in Figure 1, we consider a 2D square latticesof coupled CPWs, where the basic unit consists of a CPWcoupled to a single CBJJ, and each basic unit is connectedby a central coupler. In our case, individual central couplercould serve as a quantum transducer to realize the photon-ic tunneling process between pairs of CPWs, namely, therealization of transferring photonic states between the adja-cent CPWs in a tunable way. We can conceive the centralcoupler as the following quantum systems, such as an activenon-reciprocal devices as proposed in ref. [34], a Josephsonring circuit [32], or a capacitive coupling element in ref. [35].The Hamiltonian of the microwave-driven CPW (consistingof a narrow center conductor and two nearby lateral groundplanes) is HCPW = ωra†a (in units of � = 1), where a (a†)is the annihilation (creation) operator of the full-wave mode,

and the frequency of this mode is slightly renormalized bythe wiring and coupling capacitor as ωr/2π ≈ ε/(

√FrCr)

with ε = (1 − ε1 − ε2) and ε1 = 2C0/Cr and ε2 = Cc/Cr.Here Cr (Fr) is the capacitance (inductance) of CPW, andC0 (Cc) is the wiring (coupling) capacitors. The current dis-tribute and voltage inside the CPW can be described by therelations Icpw(x) = −i sin(2πx/Lr + ξ0)

√ωr/Fr(a − a†) and

Vcpw(x) = cos(2πx/Lr+ ξ0)√ωr/Cr(a†+a) with Lr the length

of CPW and ξ0 the small phase, which satisfies the condi-tion [tan ξ0]/2π = ε2. The condition that the employed CB-JJ can be simply modeled as a two-level qubit is that CBJJshould have sufficiently large anharmonicity (could preventqubit operations from exciting other transitions outside thelevels encoded in qubits), which can be obtained by makingthe CBJJ a low loss nonlinear oscillator using conventionalJosephson Junctions [31,36]. So the CBJJ can be governedby the Hamiltonian HC =

12ω10σ

z with ω10 the frequency ofthe CBJJ’S lowest energy-spacing and σz = |1〉C 〈1| − |0〉C 〈0|the Pauli spin operator. One can find that the following re-lations ω10 � 0.9ωs and ω21 � 0.81ωs could be well sat-isfied if we set the energy of |0〉C to be energy zero point.

Here ωs/2π =4√

(1 − Ib/Ic)(2√

2eIc/CJ)2 is the frequency ofplasma oscillation at the well’s bottom [37], where CJ is thejunction capacitance, Ib is the bias current, Ic is the criticalcurrent, and e is the charge of electron.

On the other hand, the CPW-CBJJ interaction Hamilto-nian reads HCC = �gcc(σ+a + σ−a†) with σ+ = |1〉C 〈0|(σ− = |0〉C 〈1|) the CBJJ’s rising (lowering) operator. In ourcase, The CPW acts as the quantum data bus to couple andcontrol the CBJJ qubit [38], and it capacitively couples to theCBJJ via the coupling capacitors Cc and junction capacitanceCJ with the coupling strength gcc = [2Cr(CJ + Cc)]−1/2ωrCc,which is a tunable qubit-resonator parameter [39] due to theflexibility of CBJJ qubit, allowing access to a wide range oftunable values for Ib and CJ.

Therefore, each basic unit in our system is governed by thetotal Hamiltonian H0

i = HCPW+HC+HCC. In what follows let

(a) (b)

CBJJ

Central coupler

CBJJ

Qubit

|2>|1>|0>

ω21

ω10 gcc

Cc

IcIb CJ

Figure 1 (Color online) (a) Schematic circuit for the resonator lattice,where each resonator consists of a narrow center conductor and two near-by lateral ground planes, and each resonator is coupled to a single CBJJqubit. (b) The configuration and level scheme of a CBJJ qubit. When thebias current Ib is driven close to the critical bias Ic, there exist only a fewbound states |n〉C with energy En in each washboard well.

2094 Liu Y M, et al. Sci China-Phys Mech Astron November (2014) Vol. 57 No. 11

us study the overall properties of this hybrid system. Consid-ering the nonlocal microwave PH between adjacent CPWs,the Hamiltonian has the form

H =∑

j

H0j +∑〈 j,k〉

γ〈 j,k〉a+j aγ −∑

j

μ jN j, (1)

by adding the chemical potential term at each site, whereμ j is the chemical potential at the jth site, and the PH ratesbetween jth and kth resonators can be expressed by γ〈 j,k〉 =2Z0C〈 j,k〉(ωr + δ j)(ωr + δ j) with Z0 characteristic impedance,which are set by the mutual capacitance C〈 j,k〉 between thedifferent CPWs. The PH rates γ〈 j,k〉 could be simplified toγ = 2Z0Cω2

r for the nearest-neighbor resonators, and γ〈 j,k〉= 0 for other resonator pairs if the frequency shift δ j resultingfrom random disorder could be negligible [26]. The Jaynes–Cummings (JC) type interaction between CBJJ and CPWhas a global U(1) symmetry, so that the total polariton num-ber Nj = a+j a j + σ

+j σ−j with σi the spin-1/2 Pauli operators

(i = x, y,±), is always conserved. One can find that the eigen-states of the Hamiltonian H0

j could be denoted by the dressed

states |±, n〉 with the eigenvalues E|±,n〉 = nωr − Δ2 ± ζ(n),

where ζ(n) =√

4ng2 + Δ2/2, as shown in Figure 2(a). HereΔ = ωr − ω0 is the frequency detuning between the CPWand CBJJ, n is the photon number in the CPW and |±〉 =(|0〉 ± |1〉)/√2. Note that the present system is similar to theBose–Hubbard model (BHM) [40], whereas the obvious dif-ference is that the conserved particles is the polariton ratherthan the pure bosons in BHM, and the effective on-site repul-sion U±(n) = E|±,n+1〉 −E|±,n〉 decreases with the growth of thenumber of photons, and U±(n) is infinitely close to zero whenthe values of n are very big, as shown in Figure 2(b), whichis a constant in BHM [1]. In our case, the on-site dynam-ics is governed by the interaction of JC-type, which enablesthe inter-conversion of qubit excitations and photons, andprovides the effective on-site repulsion. Meantime, pairs of

0 10 20 30 40 50n

−20 −10 0 10 20

E|±

,n>

5

0

−5

U±

0.4

0.2

0.0

−0.2

−0.4

(a) (b)

Δ/g

Figure 2 (Color online) (a) Eigenspectrum for a single CBJJ in a CPW, asa function of the CBJJ-CPW detuning Δ, where the solid (dotted) lines fromthe top to bottom denote the E|+,n〉 (E|−,n〉) in the case of Δ = −2g, 0, and2g, respectively, where g = gcc = 1. (b) The dependence of effective on-siterepulsion U± on the photon number n under the different detuning Δ, wherethe solid lines from the top to bottom denote U+ in the case of n = 1, 3, 5, 7,and 9, respectively, and the dotted lines from the top to bottom denote U− inthe case of n = 9, 7, 5, 3, and 1, respectively, where g = gcc = 1.

CPWs are coupled by the two-site Hubbard model via a one-photon PH process, which leads to the nonlocal photonic tun-neling among the different sites in this 2D configuration.

Here, one can begin to study the emergence of correlatedbehavior by calculating the corresponding order parametersψ = 〈a j〉 to distinguish the phase boundary. Note that theorder parameters ψ are set to be real in our case. Under thedecoupling approximation a+j ak = 〈a+j 〉ak +a+j 〈ak〉− 〈a+j 〉〈ak〉,and using the mean field (MF) method [41], the resulting MFHamiltonian can be written as:

Hmf =∑

j

[H0j − zγψ(a+j + a j) + zγψ2 − μ jN j], (2)

where z = 4 is the number of nearest neighbors in our system,which can be easily deduced from Figure 1. By calculatingthe system ground states corresponding to different valuesof γ and μ, we can obtain a static phase diagram in the {γ,μ} plane, and distinguish the MIP-SFP phase boundary in anintuitive manner. Varying the values of γ from the weak PHregime (γ � g) to the strong PH regime (γ g) under theresonant/detuning case, we can obtain the mean field phasediagram and phase boundary through a numerical simulationin a complete parameter space, as shown in Figure 3. At firstglance, the QPT in such a hybrid system seems surprisingbecause the photons do not normally interact with each otherwith any appreciable strength. However, one can find that,by varying the ratios of γ/g, the nonlinearity generated fromthe corresponding photon blockade effect [42] can be tunedand hence the system can be driven through the localization-delocalization phase transition, which is also accompanied bya transition of the excitations from polaritonic to photonic. Asshown in Figure 3, the region with ψ = 0 corresponds to thestable and incompressible MIP lobes. We find that this dis-tinct MIP region is confined to the weak PH regime (γ � g),

(a) (b)

(c) (d)

0.0

−0.4

−0.8

−1.2

−1.6

−2.0

(μ−ω

)/g

10−4 10−3 10−2 10−1 100 101

γ/g

0.0

−0.4

−0.8

−1.2

−1.6

−2.0

(μ−ω

)/g

10−4 10−3 10−2 10−1 100 101

γ/g0.0

−0.4

−0.8

−1.2

−1.6

−2.0

(μ−ω

)/g

10−4 10−3 10−2 10−1 100 101

γ/g

0.0

−0.4

−0.8

−1.2

−1.6

−2.0

(μ−ω

)/g

10−4 10−3 10−2 10−1 100 101

γ/g

0

1

2

3

4

Figure 3 (Color online) The order parameter ψ as a function of the chem-ical potential μ and the photon hopping rate γ for the resonance cases (a)Δ = 0, and different detuning case (b) Δ = 5g, (c) Δ = −5g, where theparameter is g = 1. The corresponding phase boundaries are plotted in (d),where the solid, dashed, and dot-dashed lines denote Δ = 0, 5g, and −5g,respectively. Here g = gcc = 1.

Liu Y M, et al. Sci China-Phys Mech Astron November (2014) Vol. 57 No. 11 2095

and SFP region exists in the strong PH regime (γ g). An-other feature is that the sizes of the MIP lobes vary with thevalues of detuning Δ, where the largest MIP lobes could befound only in the case of resonance Δ = 0. These rich re-sults can be understood as follows. In each MIP, the strongCBJJ-CPW interaction leads to an effective large polariton-polariton repulsion, which makes the number of excitationsper site constant thus with zero variance. On the contrary,the strong PH process favors the behavior of photonic delo-calization and results in the phenomena of condensation ofthe particles into the zero-momentum state. Therefore, thecompressible SFP with the stable ground state at each site(ψ � 0) corresponds to a coherent state of excitation. As a re-sult, a nonzero variance of the total excitation number exists,that is to say, the excitations at each site in such a CPW latticesystem are uncertain. In general, the boundary between theMIP (ψ = 0) and SFP (ψ � 0) denotes the position where aphotonic localization-delocalization QPT in this system willtake place for the lowest energy state.

The physical picture behind is that the JC-type interactionbetween CBJJ and CPW provides the effective on-site repul-sion, and the PH is represented by the nonlocal coupling be-tween the adjacent CPWs. The dynamics can be described asthe following process: the initial self-trapped photon popula-tion (localized regime) is driven to another different regimewhere it becomes photon population imbalance coherentlyoscillating between two CPWs (delocalized regime) when thePH rate is increased beyond a critical value. Therefore, thephase boundary primarily depends on the ratio of the PH rateto the on-site repulsion rate, and the interplay between po-lariton delocalization and on-site repulsive interaction leadsto the MIP-SFP transition. We emphasize that the key PHrates are also tunable experimental parameters by varying themutual capacitance C〈 j,k〉 between resonator ends, which willbe addressed in the following parts. Although experimentalprocedures for realizing such a suitably engineered chemicalpotential need to be developed, the proposed system allowsthe monitoring of photonic localization-delocalization transi-tions by use of currently available experimental technology.

An in-depth study of the phase diagram and a full under-standing of quantum criticality in the presence of dissipativeeffects are important. Here we also demonstrate the possi-bility of monitoring QPT of light even in a nonequilibriumscenario, including dissipative effects. Using the quantummaster equation, the MF Hamiltonian Hmf with dissipationhas the following form

ρ(t) = i�[ρ(t),Hmf

]+Lρ(t),

Lρ(t) =∑

j

[Γ

2

(2σ−j ρ(t)σ+j − σ+jσ−j ρ(t) − ρ(t)σ+jσ

−j

)

+κ

2

(2a jρ(t)a+j − a+j a jρ(t) − ρ(t)a+j a j

)], (3)

where ρ(t) is the density matrix of the CPW-CBJJ system, andthe decay rates of the CPW are κ = 4Z2

0C2outω

3r , and Γ is the

decay rate from the |1〉C of CBJJ. The corresponding Liou-ville superoperatorL models the dissipative effects resultingfrom the Markovian environment via the local density withthe strengths κ and Γ. To solve the master equation (eq. (3)),we used a product ansatz ρ(t) = ⊗ jρ(t) j, with the reduced lo-cal density operators ρ(t) j = Tr� jρ(t), expected to hold insufficiently high spatial dimension, which allows us to in-clude mixed states. Therefore the Liouville can be construct-ed by the generic connected ρ-dependent correlation matrixas 〈ϑ(t) j〉 = 〈ϑ j〉0 + δ〈ϑ(t) j〉 by linearizing in time the masterequation, where 〈ϑ j〉0 is evaluated on the homogenous steadystate of the system. Under the different dissipative effects,we plotted the phase diagram in Figure 4, and it is seen thatthere is an obvious difference of phase boundary in the pres-ence of dissipation, compared with the closed system model,as descired in the above sections. As expected, the dissipa-tive effect reduces the final value of the order parameter ψ inthe SFP regime, and population has been lost through decaychannel, whereas in the MIP regime it brings out fluctuations,again due to the population loss [13]. One can also find thatthe growth of the dissipative rates will augment the size ofMIP. These features tell us that the dissipative effects favorthe MIP, although these dissipative effects inevitably lead tothe decrease of the exciton numbers. Therefore, the dissipa-tive effects (causing the increment of the size of MIP) actuallyinduce dynamical switching from the SFP to the MIP.

Let us discuss the experimental feasibility. Firstly, we sur-vey the relevant experimental parameters. The CPW with theinductance Fr = 55 nH and the capacitance Cr = 2 pF leadsto a full frequency with ωr/2π = 3 GHz. The CBJJ’s param-eters should be tuned to CJ = 65 pF, and Ib/Ic ≈ 0.99, whichmake ω10 ∼ [1, 10] GHz when the values of Ic are set inthe domain [8, 800] μA, and yield the CBJJ-CPW couplingrate gcc/2π ∼ [1, 143] MHz if the values of Cc vary from35 fF to 5 pF. Additionally, the dissipation parameters Γ,and κ are on the order of hundreds of kHz [28,37,43], which

(a) (b)

(c) (d)

0.0

−0.4

−0.8

−1.2

−1.6

−2.0

(μ−ω

)/g

10−4 10−3 10−2 10−1 100 101

γ/g

0.0

−0.4

−0.8

−1.2

−1.6

−2.0

(μ−ω

)/g

10−4 10−3 10−2 10−1 100 101

γ/g0.0

−0.4

−0.8

−1.2

−1.6

−2.0

(μ−ω

)/g

10−4 10−3 10−2 10−1 100 101

γ/g

0.0

−0.4

−0.8

−1.2

−1.6

−2.0

(μ−ω

)/g

10−4 10−3 10−2 10−1 100 101

γ/g

Figure 4 (Color online). The order parameter ψ in μ-k plane under thedifferent dissipative rates (a) Δ = 0, and Γ = κ = 0; (b) Δ = 0, and Γ =κ = 0.02; (c) Δ = 5g, and Γ = κ = 0.1; (d) Δ = 5g, and Γ = κ = 0.3. Theother parameter is g = gcc = 1.

2096 Liu Y M, et al. Sci China-Phys Mech Astron November (2014) Vol. 57 No. 11

are two orders smaller than the coupling rates. Furthermore,the PH rate γ depends on the tunable mutual capacitanceC〈 j,k〉 between resonator ends, i.e., in ref. [26], and the au-thors have measured devices with PH rates γ/2π from 0.8MHz to 31 MHz in lattices of resonators. Secondly, how tomeasure the quantum many-body states of light is an openquestion in quantum simulation [9]. There are several pathsto solve this problem, i.e., each CPW at the outer edges ofthe lattice, can be capacitively coupled to an additional CP-W, and this configuration will result in a photonic escape rateto the continuum. This allows us to employ vector networkanalyzers in realistic experiments to measure transmissionthrough opposite ports of the CPW lattices. Note that severalexperiments [14,44] have successfully realized the transmis-sion/reflection measurement for circuit QED arrays in smallcircuit QED arrays with one or two resonators. Conceptually,such a measurement can be executed by coupling one CPW(or several) to an external CPW, where transmitted photonscan be measured in a homodyne or heterodyne way. Addi-tionally, from the point of view of photon statistics, we can al-so employ the techniques of photon-number-dependent qubittransition [44,45], or the method of fast readout of the qubit-state through a separate low-Q resonator mode [46], to ana-lyze the two-tone spectroscopy and second-order coherencefunction of the present system. For instance, the propertiesof the system can be spectroscopically probed by driving thefirst CPW with a microwave source and detecting the outputfield of the last CPW, where the density-density correlationsfor the last CPW can be measured via the output field. Fi-nally, we briefly stress the relevant experimental progress incircuit QED lattice. It is indeed feasible to couple over 200(or 1000) CPWs with negligible disorders (on the order of afew parts in 104) in a 2D lattice using a 32 mm×32 mm sam-ple or a full two-inch wafer [13], and the random disordersof CBJJ could be expected to be even smaller than CPW dis-order with individual tunability of qubit frequencies. One ofthe primary advantages of using superconducting circuitry isthe great flexibility afforded by the nature of a nanofabricatedsystem. Nearly every parameter involved is widely tunablewith conventional lithography.

In summary, we investigate the possibility of engineeringMIP-SFP transition of light in a 2D square lattices of CP-Ws coupled to a single CBJJ. We find that the localization-delocalization transition of light results form the interplaybetween the on-site repulsion and the nonlocal tunneling,where the phase boundary in the presence of dissipative ef-fects could be obtained using the MF approach and quantummaster equation. Our work opens new perspectives in quan-tum simulation of condensed matter and many-body physicsusing a hybrid circuit QED lattices.

This work was supported by the National Science Foundation of China

(Grant Nos. 11372122, 10874122 and 11074070), and the Program for Ex-

cellent Talents at the University of Guangdong Province (Guangdong Teach-

er Letter [1010] No.79).

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